Mod-17-Cohomology of J3.2, a group of order 100465920

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • J3.2 is a group of order 100465920.
  • The group order factors as 28 · 35 · 5 · 17 · 19.
  • The group is defined by 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  • It is non-abelian.
  • It has 17-Rank 1.
  • The centre of a Sylow 17-subgroup has rank 1.
  • Its Sylow 17-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(272,51); GF(17)).

General information

  • The cohomology ring is of dimension 1 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2) · (1  −  t  +  t2  −  t3  +  t4) · (1  +  t  −  t3  −  t4  −  t5  +  t7  +  t8))
    ( − 1  +  t) · (1  +  t2) · (1  +  t4) · (1  +  t8)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 16:

  1. a_15_0, a nilpotent element of degree 15
  2. c_16_0, a Duflot element of degree 16

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_15_02

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 16 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_16_0, an element of degree 16
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 15].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(272,51); GF(17))

  1. a_15_0 → a_15_0
  2. c_16_0 → c_16_0

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_15_0 → c_2_07·a_1_0, an element of degree 15
  2. c_16_0 → c_2_08, an element of degree 16

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010